A unique technique for analytical solution of 2-D dual phase lag bio-heat transfer problem with generalized time-dependent boundary conditions

Abstract

The eigenfunction-based solutions for various heat conduction models (Fourier and non-Fourier) exhibit mismatch at the boundaries due to the discrepancy between the boundary conditions (BCs) and the corresponding eigenfunction profiles. This mismatch leads to spurious oscillations and erroneous solutions near the boundaries. Recently developed finite integral transform (FIT) based analytical solution of the dual phase lag (DPL) bio-heat transfer problem with generalized time-dependent BCs in a 2-D planar skin tissue phantom also encounters the similar mismatch, especially for Dirichlet BCs. The present study aims at removing such mismatch from eigenfunction-based solutions, by the homogenization of generalized time-dependent BCs which is essentially carried out by subtracting an auxiliary function from the temperature of the domain of interest. This novel treatment yields a modified problem in terms of the modified temperature with homogeneous BCs and contains a modified source. For this, the above-mentioned auxiliary functions need to satisfy a set of conditions at the boundaries. However, the functions satisfying such conditions may not be unique. Therefore, an additional condition (pseudo-steady state condition) is imposed with a view to obtaining relatively simple, unique auxiliary functions. The current study also explores the homogenization process for 2-D bio-heat transfer problem for the tissue phantom in conjunction with the use of orthogonal eigenfunction expansion method (OEEM). The present developed solution approach is demonstrated using two test problems comprising of (a) constant surface temperature and (b) sinusoidal heat flux on the surface, respectively. In case of constant surface temperature, the present approach gets rid of large spurious oscillations near the boundary. In case of surface heat flux, temperature prediction by the present approach shows good agreement in the domain compare to that predicted by the FIT approach. Besides that, the flux distribution predicted on the boundary by present approach is a realistic non-zero value, whereas, the FIT approach produces zero heat flux for the same applied non-zero heat flux at the boundary. Moreover, for both the test problems, the developed novel approach facilitates significant reduction in number of terms in the series summation as compared to the corresponding FIT solutions, for the same level of accuracy.